4.7. Springs and Conservation of Energy. Conservation of Mechanical Energy

Transcription

1 Springs and Conservation of Energy Most drivers try to avoid collisions, but not at a deolition derby like the one shown in Figure 1. The point of a deolition derby is to crash your car into as any other cars as possible. Each car tries to daage the other cars so uch that they will stop working. The harder the crash, the ore daage you are likely to do. The last car running is the winner. 4.7 Figure 1 In a deolition derby, the cars can be crashed, but the drivers ust reain safe. How can the drivers of deolition cars avoid serious injury? What types of safety equipent do they use? Like ost drivers, they wear seat belts to hold theselves securely in their seat. They have shoulder straps to prevent lurching forward. Padding inside the driver s-side door ight provide cushioning fro side ipacts. Most cars on the road, however, have safety features that are issing or uniportant in deolition cars. Cars you ride in probably have airbags to cushion the passengers during a crash. They ay have anti-lock brakes or other coputercontrolled systes that act during eergency situations. In this section, you will explore the physics behind safety equipent and other systes in which energy is stored and transfored. CAREER LINK Conservation of Mechanical Energy Systes that ake a car safe use either springs or elastic aterials, so they have elastic potential energy. You have read about the conservation of energy in an isolated syste. The law of conservation of energy includes elastic potential energy: Energy is neither created nor destroyed in an isolated syste, but it can be transfored between kinetic energy, gravitational potential energy, elastic potential energy, and other fors of energy. In this section, we will explore interactions of systes in which echanical energy is conserved; that is, the total aount of kinetic, gravitational potential, and elastic potential energy reains constant. Energy losses due to effects such as friction, air resistance, theral energy, and sound can be ignored as negligible. Unit TASK BOOKMARK You can apply what you learn about springs and conservation of energy to the Unit Task on page 270. NEL 4.7 Springs and Conservation of Energy 201

2 Suppose, for exaple, a student jups up fro a diving board. Assuing air friction is negligible, the echanical energy of the diver will be conserved. Use the diving board as the reference point, y 5 0, for easuring the gravitational potential energy. At the axiu height h above the diving board, the diver has only gravitational potential energy equal to gdy. He then starts to fall toward the board, gaining kinetic energy because of his otion. The diver s distance above the reference point is decreasing, so his gravitational potential energy is decreasing. His total echanical energy does not change. Halfway to the board, his gravitational potential energy has decreased by half, so it exactly equals his kinetic energy. At the oent just before the diver hits the diving board, the gravitational potential energy is zero, and he has kinetic energy that equals his starting gravitational potential energy. Conservation of echanical energy still applies after the diver hits the diving board. He applies a downward force on the board, displacing it a distance x. This work transfers the diver s kinetic energy to the board. The energy is stored in the board as elastic potential energy. As the board dips down, the diver drops below y 5 0, so his gravitational potential energy becoes negative. The total elastic potential energy increases to offset the decrease in gravitational potential energy. In reality, soe energy is lost as friction, sound, and vibrations of the diving board. If we ignore these effects, echanical energy is conserved. You will apply the conservation of echanical energy in the following Tutorial. Tutorial 1 Applying the Law of Conservation of Energy In this Tutorial, we will analyze the transforations of gravitational potential, kinetic, and elastic potential energies of various systes. Saple Proble 1: Analyzing Energy Transforations In this proble, you will odel a collision and apply conservation of energy to analyze the outcoe. A odel car of ass 5.0 kg slides down a frictionless rap into a spring with spring constant k kn/ (Figure 2). Figure 2 k (a) The spring experiences a axiu copression of 22 c. Deterine the height of the initial release point. (b) Calculate the speed of the odel car when the spring has been copressed 15 c. (c) Deterine the axiu acceleration of the car after it hits the spring. Solution (a) Given: kg; k kn/ N/; Dx 5 22 c Required: Dy Analysis: DE g 5 gdy ; E e k 1Dx2 2 Dy Since energy is conserved, the change in potential energy of the odel car ust equal the change in elastic potential energy when the spring is copressed. Solution: If we choose the botto of the rap to be the y 5 0 reference point, the car will have no gravitational potential energy at the botto of the rap. The initial gravitational potential energy has been converted into kinetic energy. When the spring is fully copressed, the kinetic energy has been converted to elastic potential energy. Therefore, the spring s initial gravitational potential energy ust equal its fi nal elastic potential energy: E g 5 E e gdy k1dx2 2 Dy 5 k 1Dx2 2 2 g N/ kg /s 2 2 Dy one extra digit carried2 Stateent: The initial height of the odel car is 2.4. (b) Given: kg; k kn/ N/; Dx 5 15 c ; Dy Required: v 202 Chapter 4 Work and Energy NEL

3 Analysis: DE g 5 gdy ; E e k 1Dx22 ; E k v 2 Since energy is conserved, the su of the kinetic energy and the elastic potential energy when the spring is copressed ust equal the initial gravitational potential energy. Solution: The initial gravitational potential energy is E g 5 gdy kg /s E g J When the spring is copressed to Dx 5 15 c, the elastic potential energy is E e k 1Dx N/ E e J The kinetic energy when x 5 15 c ust equal the difference between the initial gravitational potential energy and the final elastic potential energy: E k 5 E g 2 E e J J E k J Finally, use E k to solve for v: 1 2 v 2 5 E k v 2 5 2E k v 5 Å 2E k 5 Å J2 5.0 kg v /s Stateent: The speed of the odel car when the spring is copressed 15 c is 5.1 /s. (c) Given: kg; k kn/ N/; Dx 5 22 c Required: a > Analysis: F > e 5 2kDx > ; F > net 5 a > The axiu acceleration occurs when the axiu force is acting, and this occurs when the spring is at the axiu copression of 22 c. Solution: Cobining Hooke s law and Newton s second law gives F > net 5 F > x a > 5 2kDx > a > 5 2 kdx> N/ kg a > /s 2 3toward rap4 Stateent: The axiu acceleration of the odel car is /s 2 directed toward the rap. Saple Proble 2: Using Elastic Potential, Kinetic, and Gravitational Potential Energies A 48 kg child bounces on a pogo stick. At the lowest point of one bounce, the copressed spring in the stick has 120 J of elastic potential energy as it copresses Assue that the pogo stick is light enough that we can ignore its ass. (a) Deterine the child s axiu height during the jup following the bounce. (b) Deterine the child s axiu speed during the jup. Solution (a) Given: 5 48 kg; E e J Required: Dy Analysis: DE g 5 gdy Solution: Choose the lowest point of the bounce as the y 5 0 reference point. At the axiu height, Dy, all elastic potential energy has converted to gravitational potential energy. E g 5 E e gdy 5 E e Dy 5 E e g 120 J kg /s 2 2 Dy Stateent: The child rises 0.26 fro the lowest point of the bounce. (b) Given: kg; E e J; h Required: v Analysis: The point of axiu speed is the point at which the spring is at its equilibriu position. At this point, all of the elastic potential energy has been converted to gravitational potential energy and kinetic energy. NEL 4.7 Springs and Conservation of Energy 203

4 DE g 5 gdy ; E k v 2 Solution: If we choose the lowest part of the bounce as the y 5 0 reference point, then at the equilibriu position, E g 5 gdy kg /s E g J 1one extra digit carried2 The kinetic energy is the difference between the initial elastic potential energy and the gravitational potential energy: Now solve for v: 1 2 v 2 5 E k v 5 Å 2E k J2 5 Å 48 kg v /s Stateent: The child s axiu speed is 1.1 /s. E k 5 E e 2 E g J J E k J 1one extra digit carried2 Saple Proble 3: A Block Pushed Up a Frictionless Rap by a Spring A block with a ass of 2.0 kg is held against a spring with spring constant 250 N/. The block copresses the spring 22 c fro its equilibriu position. After the block is released, it travels along a frictionless surface and then up a frictionless rap. The rap s angle of inclination is 30.08, as shown in Figure 3. (a) Deterine the elastic potential energy stored in the spring before the ass is released. (b) Calculate the speed of the block as it travels along the horizontal surface. (c) Deterine how far along the rap the block will travel before it stops. Figure 3 k 250 N/ d u 30.0 (a) Given: k N/; x 5 22 c Required: E e Analysis: Before the block is released, the entire echanical energy of the block spring syste is in the for of elastic potential energy stored in the copressed spring. We can use the given inforation to deterine the aount of stored energy, E e k 1Dx2 2. Solution: E e k 1Dx N/ E e J 1one extra digit carried2 Stateent: The elastic potential energy stored in the spring before the ass is released is 6.0 J. y (b) Given: E k J; kg Required: v Analysis: As the block travels along the fl at surface, all of the elastic potential energy is converted to kinetic energy. Use this to deterine the speed of the block using the equation for kinetic energy: E v 2. Solution: E v 2 2E 5 v 2 v 5 Å 2E 5 Å J kg2 v /s Stateent: The block will travel at a constant speed of 2.5 /s along the frictionless horizontal surface. (c) Given: E g J; kg; g /s 2 Required: Dy, d Analysis: As the block travels up the rap, kinetic energy is gradually converted to gravitational potential energy. When the block reaches its axiu height, all energy will be in potential for. Use this to deterine the vertical height attained, Dy, and then use trigonoetry to calculate the distance travelled along the rap, d, as shown in Figure 3. E g 5 gdy; Dy d 5 sin u Solution: Mechanical energy is conserved throughout this proble because there are no energy losses due to friction. 204 Chapter 4 Work and Energy NEL

5 The total potential energy at the top of the block s path is therefore 6.05 J. E g 5 gdy Dy 5 E g g 6.05 J kg /s 2 2 Dy Now use the sine ratio to deterine how far along the rap the block travels, d. Rearrange this equation to express d in ters of Dy and u. D y 5 d sin u d 5 D y sin u sin d Stateent: The block will travel a distance of 0.62, or 62 c, along the rap. Practice D y d 5 sin u 1. A block slides down a rap fro a fixed height and collides with a spring, copressing the spring until the block coes to rest. Copare the aount of copression in the case that the rap is frictionless to the case where the rap is not frictionless. Explain your answer. K/U T/I 2. A 3.5 kg ass slides fro a height of 2.7 down a frictionless rap into a spring. The spring copresses 26 c. Calculate the spring constant. T/I [ans: N/] 3. A 43 kg student jups on a pogo stick with spring constant 3.7 kn/. On one bounce, he copresses the stick s spring by 37 c. Calculate the axiu height he reaches on the following jup. T/I [ans: 0.60 above the copressed point] 4. A 0.35 kg branch falls fro a tree onto a trapoline. If the branch was initially 2.6 above the trapoline, and the trapoline copresses 0.14, calculate the spring constant of the trapoline. T/I [ans: N/] 5. Consider the block in Saple Proble 3. Suppose that the ass of the block is doubled at the top of its path of otion before returning down the frictionless rap. K/U T/I (a) Deterine the speed of the block as it returns along the horizontal surface. [ans: 2.5 /s] (b) Does the block have the sae kinetic energy as before along the horizontal surface? Explain your answer. (c) Will the block copress the spring twice as far as it did before? Explain your answer. If your answer is no, deterine the new value for x. (d) Suppose the coefficient of friction of the rap is Does your answer to (c) change? Explain your answer. If your answer is yes, deterine the new value for x. Perpetual Motion Machines An ideal spring would never lose energy and would continue with SHM forever, or as long as you did not disturb it. A achine that can continue to operate for an unliited aount of tie without outside help is a perpetual otion achine. To be a true perpetual otion achine, the achine ust be able to run forever without restarting or refuelling. A grandfather clock, for exaple, is not a perpetual otion achine, since you ust wind it up every now and then. perpetual otion achine a achine that can operate forever without restarting or refuelling NEL 4.7 Springs and Conservation of Energy 205

6 Figure 4 shows a device called Newton s cradle. The leftost ball has gravitational potential energy with respect to the other balls. Once released, the ball will interact with the other balls in such a way as to iitate perpetual otion. You will have the opportunity to explore the physics behind Newton s cradle in Chapter 5. Investigation Energy and Springs (page 211) You have learned about the spring constant and how the oveent of springs is related to the conservation of energy. Now you are ready to conduct an investigation to observe the conservation of energy. Figure 4 An ideal version of Newton s cradle would be a perpetual otion achine because it would never lose energy. An ideal version of Newton s cradle would never lose energy, and the cycle of falling, colliding, and rising would continue forever. It would then be a perpetual otion achine. Can you build such a achine? The answer is no. In real-world achines, soe echanical energy will always be lost fro the syste as theral energy, sound energy, or other fors of energy. This loss of energy can be useful. For exaple, the purpose of shock absorbers in cars, which we entioned in Section 4.6, is to use friction to stop the SHM of the car s springs. research This Perpetual Motion Machines Skills: Researching, Counicating SkILLS HANDBOOk A4.1 Hobbyists and serious researchers alike have attepted to design perpetual otion achines. They have not been successful, but soe of their ideas have useful applications. 1. Choose one achine, such as an analog watch, a etronoe, a fl ywheel, or a child s swing, that relies on ongoing, consistent otion to work properly. 2. Research the design principles that have been incorporated into odern versions of the achine to ake it work ore effi ciently. A. What scientifi c principles explain how the achine operates? A B. How has the design of the achine been iproved over tie? K/u C. Have iproveents been the result of the developent of new aterials, new technology, or new scientifi c discoveries? T/I D. Prepare a short presentation that suarizes your fi ndings. C WEB LINK 206 Chapter 4 Work and Energy NEL

7 Daped Haronic Motion So far, we have ignored the effect of friction on the otion of a siple haronic oscillator. The friction in a real periodic syste is referred to as daping, and the haronic otion of a syste affected by friction is called daped haronic otion. The presence of friction eans that the echanical energy of the syste will be transfored into theral energy, and the syste s otion will not continue perpetually. We can classify daped otion into three categories: underdaped, overdaped, and critically daped (Figure 5). daped haronic otion periodic otion affected by friction y (1) overdaped (2) critically daped (3) underdaped displaceent 0 t of ass y 0 Figure 5 When a daped oscillator is given a non-zero displaceent at t 5 0 and then released, it can exhibit three different types of behaviour: (1) overdaped, (2) critically daped, and (3) underdaped. Consider a pendulu. Curve 3 in Figure 5 shows how the displaceent varies with tie when the daping is weak, that is, when there is only a sall aount of friction. This curve applies to any weakly daped haronic oscillator. It describes the back-and-forth swinging of a pendulu or the otion of a ass attached to a spring on a horizontal surface when the surface is quite slippery. The syste still oscillates because the displaceent alternates between positive and negative values, but the aplitude of the oscillation gradually decreases with tie. The aplitude eventually goes to zero, but the syste undergoes any oscillations before daping brings it to rest. This type of otion is an underdaped oscillation. When quite a bit of friction exists, the oscillator is overdaped. The resulting displaceent as a function of tie in this case is shown as curve 1 in Figure 5. This type of otion happens when the ass oves through a very thick fluid, like the hydraulic fluid inside the closing echanis on any doors. If you pull the ass of an overdaped oscillator to one side and then release it, the ass oves extreely slowly back to the equilibriu. Critically daped otion falls in between the two extree cases. In underdaped otion, displaceent always passes through zero the equilibriu point at least once, and usually any ties, before the syste coes to rest. In contrast, an overdaped syste released fro rest oves just to the equilibriu point, but not beyond. In critically daped otion, displaceent falls to zero as quickly as possible without oving past the equilibriu position. Displaceent as a function of tie for the critically daped case is illustrated by curve 2 in Figure 5. These different categories of daping have different applications. For exaple, a car s shock absorbers provide daping for springs that support the car s body (Figure 6). Shock absorbers enable the tires to ove up and down over bups in the road without directly passing vibrations to the car s body or passengers. When the car hits a bup, the springs copress. To ake the ride as cofortable as possible, the shock absorbers critically dap the otion of the springs. The critically daped otion eans the body of the car returns to its original height as quickly as possible. Worn-out shock absorbers lead to underdaped otion, and the car bounces up and down ore. Overdaped shock absorbers give a soft spongy ride with poor steering response and handling. CAREER LINK shock absorber coil spring Figure 6 The shock absorbers on a car serve to dapen its coil springs. The goal is usually to have a car respond to bups in the road as a critically daped oscillator. NEL 4.7 Springs and Conservation of Energy 207

8 4.7 Review Suary For an isolated ass spring syste, the total echanical energy kinetic energy, elastic potential energy, and gravitational potential energy reains constant. A perpetual otion achine is a achine that can operate forever without restarting or refuelling. Daped haronic otion is periodic otion in which friction causes a decrease in the aplitude of otion and the total echanical energy. Questions 1. A ass hangs fro a vertical spring and is initially at rest. A person then pulls down on the ass, stretching the spring. Does the total echanical energy of this syste (the ass plus the spring) increase, decrease, or stay the sae? Explain. K/U 2. A ass rests against a spring on a horizontal, frictionless table. The spring constant is 520 N/, and the ass is 4.5 kg. The ass is pushed against the spring so that the spring is copressed by 0.35, and then it is released. Deterine the velocity of the ass when it leaves the spring. T/I 3. A toy airplane ejects its 8.4 g pilot using a spring with a spring constant of N/. The spring is initially copressed 5.2 c. T/I (a) Calculate the elastic potential energy of the copressed spring. (b) Calculate the speed of the pilot as it ejects upward fro the airplane. (c) Deterine the axiu height that the pilot will reach. 4. In a pinball gae, a copressed spring with spring constant N/ fires an 82 g pinball. The pinball first travels horizontally and then travels up an inclined plane in the achine before coing to rest. The ball rises up the rap through a vertical height of 3.4 c. Deterine the distance of the spring s copression. T/I 5. A bungee juper of ass 75 kg is standing on a platfor 53 above a river. The length of the unstretched bungee cord is 11. The spring constant of the cord is 65.5 N/. Calculate the juper s speed at 19 below the bridge on the first fall. T/I 6. A spring with a spring constant of 5.0 N/ has a 0.25 kg box attached to one end such that the box is hanging down fro the string at rest. The box is then pulled down another 14 c fro its rest position. Calculate the axiu height, the axiu speed, and the axiu acceleration of the box. T/I 7. A 0.22 kg block is dropped on a vertical spring that has a spring constant of 280 N/. The block attached to the spring copresses it by 11 c before oentarily stopping. Deterine the height fro which the block was dropped. T/I 8. A block of 1.0 kg with speed 1.0 /s hits a spring placed horizontally, as shown in Figure 7. The spring constant is N/. T/I (a) Calculate the axiu copression of the spring. (b) How far will the block travel before coing to rest? Assue that the surfaces are frictionless. Figure kg 9. A wooden box of ass 6.0 kg slides on a frictionless tabletop with a speed of 3.0 /s. It is brought to rest by a copressing spring. The spring constant is 1250 N/. T/I (a) Calculate the axiu distance the spring is copressed. (b) Deterine the speed and acceleration of the block when the spring is copressed a distance of 14 c. 10. A tennis coach uses a achine to help with tennis practice. The achine uses a copressed spring to launch tennis balls. The spring constant is 440 N/, and the spring is initially copressed 45 c. A 57 g tennis ball leaves the achine horizontally at a height of 1.2. Calculate the horizontal distance that the tennis ball can travel before hitting the ground. T/I 208 Chapter 4 Work and Energy NEL